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Bicategories of Processes
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 1997
"... The suspensionloop construction is used to define a process in a symmetric monoidal category. The algebra of such processes is that of symmetric monoidal bicategories. Processes in categories with products and in categories with sums are studied in detail, and in both cases the resulting bicate ..."
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Cited by 43 (14 self)
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The suspensionloop construction is used to define a process in a symmetric monoidal category. The algebra of such processes is that of symmetric monoidal bicategories. Processes in categories with products and in categories with sums are studied in detail, and in both cases the resulting bicategories of processes are equipped with operations called feedback. Appropriate versions of traced monoidal properties are verified for feedback, and a normal form theorem for expressions of processes is proved. Connections with existing theories of circuit design and computation are established via structure preserving homomorphisms.
On Folk Theorems
, 1980
"... this paper is to refine this definition somewhat, adapting it to the purposes of the research community in computer science. Accordingly, we shall attempt to provide a reasonable definition of or, rather, criteria for folk theorems, followed by a detailed example illustrating the ideas. The latter e ..."
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Cited by 29 (0 self)
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this paper is to refine this definition somewhat, adapting it to the purposes of the research community in computer science. Accordingly, we shall attempt to provide a reasonable definition of or, rather, criteria for folk theorems, followed by a detailed example illustrating the ideas. The latter endeavor might take one of two possible forms. We could take a piece of folklore and show that it is a theorem, or take a theorem and show that it is folklore. As an example of the first form we could have shown that the statement P NP, which is folklore, is also a theorem. However, since we have resolved to introduce no new technical material in this paper, and moreover, since researchers in our community seem to be less familiar with folklore than with theorems, Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission
EQUIVALENCES AND TRANSFORMATIONS OF REGULAR SYSTEMS  APPLICATIONS TO RECURSIVE PROGRAM SCHEMES AND GRAMMARS
, 1986
"... This work presents a unified theory of recursive program schemes, contextfree grammars, grammars on arbitrary algebraic structures and, in fact, recursive definitions of all kind by means of regular systems. The equivalences of regular systems associated with either all their solutions or their le ..."
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Cited by 27 (5 self)
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This work presents a unified theory of recursive program schemes, contextfree grammars, grammars on arbitrary algebraic structures and, in fact, recursive definitions of all kind by means of regular systems. The equivalences of regular systems associated with either all their solutions or their least solutions (in all domains of appropriate type satisfying a set of algebraic laws expressed by equations) are systematically investigated and characterized (in some cases) in terms of system transformations by folding, unfolding and rewriting according to the equational algebraic laws. Grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union, and the same questions are raised for them. We also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system. This theory applies to grammars of many kinds which generate trees, graphs, etc. We formulate some classical transformations of contextfree grammars in terms of correct transformations which only use folding, unfolding and algebraic laws and we immediately obtain their correctness.
Rational Term Rewriting
, 1998
"... . Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via terms, that is, terms over a signature extended with selfinstantiation operators. For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), ..."
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Cited by 23 (12 self)
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. Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via terms, that is, terms over a signature extended with selfinstantiation operators. For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), . . . ). Now, if we reduce a term t to s via a rewriting rule using standard notions of the theory of Term Rewriting Systems, how are the rational terms corresponding to t and to s related? We answer to this question in a satisfactory way, resorting to the definition of infinite parallel rewriting proposed in [7]. We also provide a simple, algebraic description of term rewriting through a variation of Meseguer's Rewriting Logic formalism. 1 Introduction Rational terms are possibly infinite terms with a finite set of subterms. They show up in a natural way in Theoretical Computer Science whenever some finite cyclic structures are of concern (for example data flow diagrams, cyclic te...
Algebra of Flownomials; Part 1: Binary Flownomials; Basic Theory
"... ' morphism for connecting flowgraphs are used in [CaU82] and in all of our subsequent papers on flowchart schemes and flownomials, see [Ste87a, Ste87b, CaS88a, CaS90a, CaS92]. This chapter folows Chapter B, sec. 36 of [Ste91]. The main result is based on a series of papers dealing with the al ..."
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' morphism for connecting flowgraphs are used in [CaU82] and in all of our subsequent papers on flowchart schemes and flownomials, see [Ste87a, Ste87b, CaS88a, CaS90a, CaS92]. This chapter folows Chapter B, sec. 36 of [Ste91]. The main result is based on a series of papers dealing with the algebraization of flowchart schemes, including [CaU82, BlEs85, Ste86/90, Bar87a, CaS88a, CaS90b]. With different sets of operators various algebras for flowgraphs appear in [Mil79, Parr87, CaS90b, CaS88b]. In the classical algebraic calculus for regular languages it is often the case that certain abstract semirings are used instead of the Boolean f0; 1g semiring, e.g. by using formal series with such coefficients. 5 This property is similar to the universal property of the polynomials over a ring. Chapter 6 Graph isomorphism with various constants In this chapter we extend the axiomatistion for flowgraphs modulo isomorphism to the case where more constants for generating relations are present i...
NonDeterministic Kleene Coalgebras
"... In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Miln ..."
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Cited by 14 (5 self)
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In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.
Rewriting On Cyclic Structures: Equivalence Between The Operational And The Categorical Description
, 1999
"... . We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, fo ..."
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Cited by 13 (6 self)
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. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms. R'esum'e. Nous pr'esentons une formulation cat'egorique de la r'e'ecriture des graphes cycliques des termes, bas'ee sur une variante de 2theorie alg'ebrique. Nous prouvons que cette pr'esentation est 'equivalente `a la d'efinition op'erationnelle propos'ee par Barendregt et d'autres auteurs, mais pas dons le cas des radicaux circulaires, pour lesquels nous proposons (et justifions formellem...
Algebraic state machines
 Proc. 8th Internat. Conf. Algebraic Methodology and Software Technology, AMAST 2000. LNCS 1816
, 2000
"... Abstract. We introduce the concept of an algebraic state machine. This is a state transition machine all parts of that are described by algebraic and logical means. This way we base the description of state transition systems exclusively on the concept of algebraic specifications. Also the state of ..."
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Abstract. We introduce the concept of an algebraic state machine. This is a state transition machine all parts of that are described by algebraic and logical means. This way we base the description of state transition systems exclusively on the concept of algebraic specifications. Also the state of an algebraic state machine is represented by an algebra. In particular, we describe the state spaces of the state machine by algebraic techniques, and the state transitions by special axioms called transition rules. Then we show how known concepts from algebraic specifications can be used to provide a notion of parallel composition with asynchronous interaction for algebraic state machines. As example we introduce a notion of objectoriented component and show how algebraic state machines can formalize such components. 1
Reaction and Control I. Mixing Additive and Multiplicative Network Algebras
 Logic Journal of the IGPL
, 1996
"... . This paper is included in a series aiming to contribute to the algebraic theory of distributed computation. The key problem in understanding MultiAgent Systems is to find a theory which integrates the reactive part and the control part of such systems. To this end we use the calculus of flownomi ..."
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Cited by 9 (2 self)
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. This paper is included in a series aiming to contribute to the algebraic theory of distributed computation. The key problem in understanding MultiAgent Systems is to find a theory which integrates the reactive part and the control part of such systems. To this end we use the calculus of flownomials. It is a polynomiallike calculus for representing flowgraphs and their behaviours. An `additive' interpretation of the calculus was intensively developed to study control flowcharts and finite automata. For instance, regular algebra and iteration theories are included in a unified presentation. On the other hand, a `multiplicative' interpretation of the calculus of flownomials was developed to study dataflow networks. The claim of this series of papers is that the mixture of the additive and multiplicative network algebras will contribute to the understanding of distributed computation. The role of this first paper is to present a few motivating examples. To appear in Journal of IGPL....
The algebra of stream processing functions
, 1996
"... Dataflow networks are a model of concurrent computation. They consist of a collection of concurrent asynchronous processes which communicate by sending data over FIFO channels. In this paper we study the algebraic structure of the data ow networks and base their semantics on stream processing funct ..."
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Cited by 8 (1 self)
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Dataflow networks are a model of concurrent computation. They consist of a collection of concurrent asynchronous processes which communicate by sending data over FIFO channels. In this paper we study the algebraic structure of the data ow networks and base their semantics on stream processing functions. The algebraic theory is provided by the calculus of flownomials which gives a unified presentation of regular algebra and iteration theories. The kernel of the calculus is an equational axiomatization called Basic Network Algebra (BNA) for flowgraphs modulo graph isomorphism. We show that the algebra of stream processing functions called SPF (used for deterministic networks) and the algebra of sets of stream processing functions called PSPF (used for nondeterministic networks) are BNA algebras. As a byproduct this shows that both semantic models are compositional. We also identify the additional axioms satisfied by the branching components that correspond to constants in these two algebraic theories. For the deterministic case we study in addition the coarser equivalence relation on networks given by the inputoutput behaviour and provide a correct and complete axiomatization.